Convergence with ratio/root test and power series

Okay so i missed the days in class when we went over power series and ratio/root test because of another exam. Reading my book i cant figure out how to do ratio/root test and power series. The example in the book for ratio/root test is:

Sum of: (1+n)^(n^2)
from n=1 to infiniti

The book says to take limit as n goes to infinity of an over the limit as n goes to infinity of an+1 . I don't get how this helps to determine if the series is convergent. Could someone walk me through this problem, and tell me how i know to use ratio or root test.

The book simply says the answer and doesn't include any steps as to how it got that answer

Also the power series does not make sense to me. I don't get how you identify when you use it. The example in the book is:

3/(x^2 - x - 2)

It says write it as a power series. It gives the answer but doesnt describe how to do that. And from this example i'm suppose to be able to do the rest of my homework, but i don't get how to write this as a power series. Could someone walk me through the steps of this one too?

Okay so i missed the days in class when we went over power series and ratio/root test because of another exam. Reading my book i cant figure out how to do ratio/root test and power series. The example in the book for ratio/root test is:

Sum of: (1+n)^(n^2)
from n=1 to infiniti

The book says to take limit as n goes to infinity of an over the limit as n goes to infinity of an+1 . I don't get how this helps to determine if the series is convergent. Could someone walk me through this problem, and tell me how i know to use ratio or root test.

The book simply says the answer and doesn't include any steps as to how it got that answer

Also the power series does not make sense to me. I don't get how you identify when you use it. The example in the book is:

3/(x^2 - x - 2)

It says write it as a power series. It gives the answer but doesnt describe how to do that. And from this example i'm suppose to be able to do the rest of my homework, but i don't get how to write this as a power series. Could someone walk me through the steps of this one too?

The first should actually be that if is an infinite series, then the following is true: Let . Then converges iff